<h2>Problem 229</h2>
<div style="color:#666;font-size:80%;">24 January 2009</div><br />
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<p>Consider the number 3600. It is very special, because</p>
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3600 = 48<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + &nbsp;&nbsp; &nbsp;36<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /><br /><br />
3600 = 20<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 2<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />40<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /><br /><br />
3600 = 30<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 3<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />30<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /><br /><br />
3600 = 45<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 7<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />15<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /><br /><br />
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<p>Similarly, we find that 88201 = 99<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 280<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 287<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 2<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />54<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 283<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 3<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />52<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 197<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 7<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />84<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />.</p>

<p>In 1747, Euler proved which numbers are representable as a sum of two squares.
We are interested in the numbers <var>n</var> which admit representations of all of the following four types:</p>
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<var>n</var> = <var>a<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /></var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + &nbsp; <var>b<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /></var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /><br /><br />
<var>n</var> = <var>a<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /></var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 2 <var>b<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /></var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /><br /><br />
<var>n</var> = <var>a<img src="" style="display:none;" alt="_(" /><sub>3</sub><img src="" style="display:none;" alt=")" /></var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 3 <var>b<img src="" style="display:none;" alt="_(" /><sub>3</sub><img src="" style="display:none;" alt=")" /></var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /><br /><br />
<var>n</var> = <var>a<img src="" style="display:none;" alt="_(" /><sub>7</sub><img src="" style="display:none;" alt=")" /></var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 7 <var>b<img src="" style="display:none;" alt="_(" /><sub>7</sub><img src="" style="display:none;" alt=")" /></var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />,
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<p>where the <var>a</var><img src="" style="display:none;" alt="_(" /><sub><var>k</var></sub><img src="" style="display:none;" alt=")" /> and <var>b</var><img src="" style="display:none;" alt="_(" /><sub><var>k</var></sub><img src="" style="display:none;" alt=")" /> are positive integers.</p>

<p>There are 75373 such numbers that do not exceed 10<img src="" style="display:none;" alt="^(" /><sup>7</sup><img src="" style="display:none;" alt=")" />.<br />

How many such numbers are there that do not exceed 2<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />10<img src="" style="display:none;" alt="^(" /><sup>9</sup><img src="" style="display:none;" alt=")" />?</p>
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